1 Model

Loss curves are a standard actuarial technique for helping insurance companies assess the amount of reserve capital they need to keep on hand to cover claims from a line of business. Claims made and reported for a given accounting period are tracked seperately over time so that more recent periods with less claim development can use the behaviour of policies from earlier periods of time to help predict what total claims will be.

Total claim amounts from a simple accounting period are usually laid out in a single line with each column showing the total claim amount after that period of time. Subsequent accounting periods have less development and so the data takes a triangular shape, hence the term ‘loss triangles’. Using previous patterns, data in the upper part of the triangle can be used to predict values in the unknown lower triangle.

The ChainLadder package provides functionality to generate and use these loss triangle.

In this case study, we take a related but different approach: we model the growth of the losses in each accounting period as an increasing function of time, and use the model to estimate the parameters which determine the shape and form of this growth. We also use the sampler to estimate the values of the “ultimate loss ratio”, i.e. the ratio of the total claims on an accounting period to the total premium received to write those policies.

1.1 Overview

We will work with two different functional forms for the growth behaviour of the loss curves: a ‘Weibull’ model and a ‘loglogistic’ model:

\[ g(t \, ; \; \theta, \omega) = \frac{t^\omega}{t^\omega + \theta^\omega} \;\; (\text{Weibull}) \]

\[ g(t \, ; \; \theta, \omega) = 1 - \exp\left(-\left(\frac{t}{\theta}\right)^\omega\right) \;\; (\text{Log-logistic}) \]

2 Load Data

We will first look at the car insurance loss data from casact.org

### File was downloaded from http://www.casact.org/research/reserve_data/ppauto_pos.csv

ppauto_dt <- fread("data/ppauto_pos.csv")

setnames(ppauto_dt, tolower(names(ppauto_dt)))

head(ppauto_dt)
##    grcode                  grname accidentyear developmentyear developmentlag
## 1:     43 IDS Property Cas Ins Co         1988            1988              1
## 2:     43 IDS Property Cas Ins Co         1988            1989              2
## 3:     43 IDS Property Cas Ins Co         1988            1990              3
## 4:     43 IDS Property Cas Ins Co         1988            1991              4
## 5:     43 IDS Property Cas Ins Co         1988            1992              5
## 6:     43 IDS Property Cas Ins Co         1988            1993              6
##    incurloss_b cumpaidloss_b bulkloss_b earnedpremdir_b earnedpremceded_b
## 1:         607           133        226             957                62
## 2:         647           333        129             957                62
## 3:         582           431         38             957                62
## 4:         598           570         19             957                62
## 5:         614           615          0             957                62
## 6:         615           615          0             957                62
##    earnedpremnet_b single postedreserve97_b
## 1:             895      0             73044
## 2:             895      0             73044
## 3:             895      0             73044
## 4:             895      0             73044
## 5:             895      0             73044
## 6:             895      0             73044

In terms of modeling the work, we want to ensure the data we work with is a snapshot in time. We

use_grcode <- c(43,353,388,620)

carrier_snapshot_dt <- ppauto_dt %>%
    filter(grcode %in% use_grcode
          ,developmentyear < 1998) %>%
    mutate(acc_year   = accidentyear
          ,dev_lag    = developmentlag
          ,premium    = earnedpremdir_b
          ,cum_loss   = cumpaidloss_b
          ,loss_ratio = cum_loss / premium) %>%
    select(grcode, acc_year, dev_lag, premium, cum_loss, loss_ratio)

We are looking at four insurers with the GRCODEs above. Before we proceed with any analysis, we first plot the data, grouping the loss curves by accounting year and faceting by carrier.

ggplot(carrier_snapshot_dt) +
    geom_line(aes(x = dev_lag, y = loss_ratio, colour = as.character(acc_year))
             ,size = 0.3) +
    expand_limits(y = c(0,1)) +
    facet_wrap(~grcode) +
    xlab('Development Time') +
    ylab('Loss Ratio') +
    ggtitle('Snapshot of Loss Curves for 10 Years of\nPrivate Passenger Auto Insurance for Single Organisation') +
    guides(colour = guide_legend(title = 'Cohort Year'))

We look at the chain ladder of the data, rather than looking at the loss ratios we just look at the dollar amounts of the losses.

snapshot_dt <- carrier_snapshot_dt %>%
    filter(grcode %in% use_grcode[1])

snapshot_dt %>%
    select(acc_year, dev_lag, premium, cum_loss) %>%
    spread(dev_lag, cum_loss) %>%
    print
##     acc_year premium     1     2     3     4     5     6     7    8    9  10
##  1:     1988     957   133   333   431   570   615   615   615  614  614 614
##  2:     1989    3695   934  1746  2365  2579  2763  2966  2940 2978 2978  NA
##  3:     1990    6138  2030  4864  6880  8087  8595  8743  8763 8762   NA  NA
##  4:     1991   17533  4537 11527 15123 16656 17321 18076 18308   NA   NA  NA
##  5:     1992   29341  7564 16061 22465 25204 26517 27124    NA   NA   NA  NA
##  6:     1993   37194  8343 19900 26732 30079 31249    NA    NA   NA   NA  NA
##  7:     1994   46095 12565 26922 33867 38338    NA    NA    NA   NA   NA  NA
##  8:     1995   51512 13437 26012 31677    NA    NA    NA    NA   NA   NA  NA
##  9:     1996   52481 12604 23446    NA    NA    NA    NA    NA   NA   NA  NA
## 10:     1997   56978 12292    NA    NA    NA    NA    NA    NA   NA   NA  NA

We also look at the loss ratios in a similar fashion

snapshot_dt %>%
    select(acc_year, dev_lag, premium, loss_ratio) %>%
    spread(dev_lag, loss_ratio) %>%
    print(digits = 2)
##     acc_year premium    1    2    3    4    5    6    7    8    9   10
##  1:     1988     957 0.14 0.35 0.45 0.60 0.64 0.64 0.64 0.64 0.64 0.64
##  2:     1989    3695 0.25 0.47 0.64 0.70 0.75 0.80 0.80 0.81 0.81   NA
##  3:     1990    6138 0.33 0.79 1.12 1.32 1.40 1.42 1.43 1.43   NA   NA
##  4:     1991   17533 0.26 0.66 0.86 0.95 0.99 1.03 1.04   NA   NA   NA
##  5:     1992   29341 0.26 0.55 0.77 0.86 0.90 0.92   NA   NA   NA   NA
##  6:     1993   37194 0.22 0.54 0.72 0.81 0.84   NA   NA   NA   NA   NA
##  7:     1994   46095 0.27 0.58 0.73 0.83   NA   NA   NA   NA   NA   NA
##  8:     1995   51512 0.26 0.50 0.61   NA   NA   NA   NA   NA   NA   NA
##  9:     1996   52481 0.24 0.45   NA   NA   NA   NA   NA   NA   NA   NA
## 10:     1997   56978 0.22   NA   NA   NA   NA   NA   NA   NA   NA   NA

2.1 Quick Data Exploration

We are working with the loss ratio, so we recreate the chain ladder format but look at loss ratios instead of dollar losses.

ggplot() +
    geom_line(aes(x = dev_lag, y = loss_ratio, colour = as.character(acc_year))
             ,data = snapshot_dt
             ,size = 0.3) +
    expand_limits(y = 0) +
    xlab('Development Time') +
    ylab('Loss Ratio') +
    guides(colour = guide_legend(title = 'Cohort Year'))

3 Stan Model

3.1 Configure Data

We want to only use the data at a given snapshot, so we choose all data current to 1998. Thus, we have 10 years of development for our first 1988 cohort, and one less for each subsequent year. Our final cohort for 1997 has only a single year of development

cohort_dt <- snapshot_dt %>%
    select(acc_year, premium) %>%
    unique() %>%
    mutate(cohort_id = 1:n())

usedata_dt <- snapshot_dt

cohort_maxtime <- usedata_dt %>%
    group_by(acc_year) %>%
    summarise(maxtime = max(dev_lag)) %>%
    arrange(acc_year) %>%
    .[[2]]

cohort_premium <- usedata_dt %>%
    group_by(acc_year) %>%
    summarise(premium = unique(premium)) %>%
    .[[2]]

t_values <- usedata_dt %>%
    select(dev_lag) %>%
    arrange(dev_lag) %>%
    unique %>%
    .[[1]]

standata_lst <- list(growthmodel_id = 1   # Use weibull rather than loglogistic
                    ,n_data         = usedata_dt %>% nrow
                    ,n_time         = usedata_dt %>% select(dev_lag)  %>% unique %>% nrow
                    ,n_cohort       = usedata_dt %>% select(acc_year) %>% unique %>% nrow
                    ,cohort_id      = get_character_index(usedata_dt$acc_year)
                    ,cohort_maxtime = cohort_maxtime
                    ,t_value        = t_values
                    ,t_idx          = get_character_index(usedata_dt$dev_lag)
                    ,premium        = cohort_premium
                    ,loss           = usedata_dt$cum_loss
                    )

The full Stan file is shown below:

functions {
    real growth_factor_weibull(real t, real omega, real theta) {
        real factor;

        factor = 1 - exp(-(t/theta)^omega);

        return(factor);
    }

    real growth_factor_loglogistic(real t, real omega, real theta) {
        real factor;

        factor = ((t^omega) / (t^omega + theta^omega));

        return(factor);
    }
}

data {
    int<lower=0,upper=1> growthmodel_id;

    int n_data;
    int n_time;
    int n_cohort;

    int cohort_id[n_data];
    int t_idx[n_data];

    int cohort_maxtime[n_cohort];

    real<lower=0> t_value[n_time];

    real premium[n_cohort];
    real loss[n_data];
}

parameters {
    real<lower=0> omega;
    real<lower=0> theta;

    real<lower=0> LR[n_cohort];

    real mu_LR;
    real<lower=0> sd_LR;

    real<lower=0> loss_sd;
}

transformed parameters {
    real gf[n_time];
    real loss_mean[n_cohort, n_time];

    for(i in 1:n_time) {
        if(growthmodel_id == 1) {
            gf[i] = growth_factor_weibull    (t_value[i], omega, theta);
        } else {
            gf[i] = growth_factor_loglogistic(t_value[i], omega, theta);
        }
    }

    for(i in 1:n_data) {
        loss_mean[cohort_id[i], t_idx[i]] = LR[cohort_id[i]] * premium[cohort_id[i]] * gf[t_idx[i]];
    }
}

model {
    mu_LR ~ normal(0, 0.5);
    sd_LR ~ lognormal(0, 0.5);

    LR ~ lognormal(mu_LR, sd_LR);

    loss_sd ~ lognormal(0, 0.7);

    omega ~ lognormal(0, 1);
    theta ~ lognormal(0, 1);

    for(i in 1:n_data) {
        loss[i] ~ normal(loss_mean[cohort_id[i], t_idx[i]], premium[cohort_id[i]] * loss_sd);
    }
}


generated quantities {
    real mu_LR_exp;
    real<lower=0> loss_sample[n_cohort, n_time];
    real<lower=0> loss_prediction[n_cohort, n_time];
    real<lower=0> step_ratio[n_cohort, n_time];

    for(i in 1:n_cohort) {
        for(j in 1:n_time) {
            loss_sample[i, j] = LR[i] * premium[i] * gf[t_idx[j]];
            step_ratio [i, j] = 1.0;
        }
    }

    mu_LR_exp = exp(mu_LR);

    for(i in 1:n_data) {
        loss_prediction[cohort_id[i], t_idx[i]] = loss[i];
    }

    for(i in 1:n_cohort) {
        for(j in 2:n_time) {
            step_ratio[i, j] = gf[t_idx[j]] / gf[t_idx[j-1]];
        }
    }

    for(i in 1:n_cohort) {
        for(j in (cohort_maxtime[i]+1):n_time) {
            loss_prediction[i,j] = loss_prediction[i,j-1] * step_ratio[i,j];
        }
    }
}

Need to discuss a number of issues with the above file:

  • Use of functions to select between the Weibull and Loglogistic
  • Use of lognormal to ensure positive values on LR and stddevs
  • Use of normal, not lognormal, for mu_LR as it feeds into a lognormal
  • Use of generated quantities
lc_1_stanfit <- stan(file   = 'losscurves_single.stan'
                    ,data   = standata_lst
                    ,iter   = 500
                    ,chains = 8
                    )

The Stan sample contains no divergent transitions, so that is a good start.

It is always worth checking convergence of the model by checking the \(\hat{R}\) and ensuring it is less than about 1.1

# Plot of convergence statistics
lc_1_draws       <- extract(lc_1_stanfit, permuted = FALSE, inc_warmup = TRUE)
lc_1_monitor_tbl <- as.data.frame(monitor(lc_1_draws, print = FALSE))
lc_1_monitor_tbl <- lc_1_monitor_tbl %>%
    mutate(variable  = rownames(lc_1_monitor_tbl)
          ,parameter = gsub("\\[.*]", "", variable)
           )

ggplot(lc_1_monitor_tbl) +
    aes(x = parameter, y = Rhat, color = parameter) +
    geom_jitter(height = 0, width = 0.2, show.legend = FALSE) +
    geom_hline(aes(yintercept = 1), size = 0.5) +
    theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
    ylab(expression(hat(italic(R))))
## Warning: Removed 110 rows containing missing values (geom_point).

The \(\hat{R}\) values for the parameter appear to be in or around 1, so that is encouraging.

Finally, we look at some of the n_eff variable also

ggplot(lc_1_monitor_tbl) +
    aes(x = parameter, y = n_eff, color = parameter) +
    geom_jitter(height = 0, width = 0.2, show.legend = FALSE) +
    expand_limits(y = 0) +
    theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
    xlab("Parameter") +
    ylab(paste0("Effective Sample Count (n_eff)"))

param_root <- c("omega", "theta", "LR", "gf", "loss_sd")

use_vars <- lc_1_monitor_tbl %>%
    filter(parameter %in% param_root) %>%
    .[["variable"]]

Check the traces. There are a large number of parameters in this model fit, so we break them up into groups. First we look at omega, theta and LR

rstan::traceplot(lc_1_stanfit, pars = c("omega", "theta", "LR")) +
    theme(axis.text.x = element_text(angle = 90, vjust = 0.5))

Now we look at the traces for gf and loss_sd.

rstan::traceplot(lc_1_stanfit, pars = c("gf", "loss_sd")) +
    theme(axis.text.x = element_text(angle = 90, vjust = 0.5))

Now we check the 50% credibility intervals of the parameters

plotdata_dt <- lc_1_monitor_tbl %>%
    filter(variable %in% use_vars) %>%
    select(mean, `25%`, `50%`, `75%`) %>%
    mutate(variable = factor(use_vars, levels = use_vars))

ggplot(plotdata_dt) +
    geom_point(aes(x = variable, y = mean)) +
    geom_errorbar(aes(x = variable, ymin = `25%`, ymax = `75%`), width = 0) +
    expand_limits(y = 0) +
    theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
    xlab("Parameter") +
    ylab("Value")

Now that we have our fit we can start looking at some plots for it. First we look at some very simple sanity-check plots. We look at the full development of an accounting year and see how our fit is doing at tracking that.

fitted_curves_tbl <- extract(lc_1_stanfit)$loss_sample[,1,] %>%
    as_data_frame() %>%
    mutate(iter = 1:n()) %>%
    gather("timelbl", "value", -iter) %>%
    mutate(time = gsub("V", "", timelbl) %>% as.numeric())

ggplot(snapshot_dt %>% filter(acc_year == 1988)) +
    geom_line (aes(x = time, y = value, group = iter)
              ,data = fitted_curves_tbl, alpha = 0.01) +
    geom_line (aes(x = dev_lag, y = cum_loss), colour = 'red') +
    geom_point(aes(x = dev_lag, y = cum_loss), colour = 'blue') +
    expand_limits(y = 0) +
    scale_y_continuous(labels = scales::dollar) +
    ggtitle("Plot of 1988 Year Loss Development Against Posterior Distribution") +
    xlab("Time") +
    ylab("Loss")

Seems to be doing a reasonable job. Now we use it to predict forward. We start with 1993, where we have development of five years.

predict_cone_tbl <- extract(lc_1_stanfit)$loss_prediction[,6,] %>%
    as_data_frame() %>%
    mutate(iter = 1:n()) %>%
    gather("timelbl", "value", -iter) %>%
    mutate(time = gsub("V", "", timelbl) %>% as.numeric())

plot_predict <- ggplot(ppauto_dt %>% filter(grcode == 43, accidentyear == 1993)) +
    geom_line (aes(x = time, y = value, group = iter)
              ,data = predict_cone_tbl, alpha = 0.01) +
    geom_line (aes(x = developmentlag, y = cumpaidloss_b), colour = 'red') +
    geom_point(aes(x = developmentlag, y = cumpaidloss_b), colour = 'blue') +
    expand_limits(y = 0) +
    scale_y_continuous(labels = scales::dollar) +
    ggtitle("Plot of 1993 Year Loss Prediction") +
    xlab("Time") +
    ylab("Loss")

plot(plot_predict)

We now look at a later year, with less claim development.

predict_cone_tbl <- extract(lc_1_stanfit)$loss_prediction[,8,] %>%
    as_data_frame() %>%
    mutate(iter = 1:n()) %>%
    gather("timelbl", "value", -iter) %>%
    mutate(time = gsub("V", "", timelbl) %>% as.numeric())

plot_predict <- ggplot(ppauto_dt %>% filter(grcode == 43, accidentyear == 1995)) +
    geom_line (aes(x = time, y = value, group = iter)
              ,data = predict_cone_tbl, alpha = 0.01) +
    geom_line (aes(x = developmentlag, y = cumpaidloss_b), colour = 'red') +
    geom_point(aes(x = developmentlag, y = cumpaidloss_b), colour = 'blue') +
    expand_limits(y = 0) +
    scale_y_continuous(labels = scales::dollar) +
    ggtitle("Plot of 1995 Year Loss Prediction") +
    xlab("Time") +
    ylab("Loss")

plot(plot_predict)

3.2 SolvencyII and the 99.5 Percentile

4 Multiple Insurers

stanfile <- 'losscurves_multiple.stan'

grcodes    <- ppauto_dt[, .(pos_prem = all(earnedpremdir_b > 0)), by = grcode][pos_prem == TRUE, grcode]
use_grcode <- grcodes[1:15]

multi_dt <- ppauto_dt[developmentyear < 1998 &
                      grcode %in% use_grcode, .(grcode    = grcode
                                               ,accyear   = accidentyear
                                               ,premium   = earnedpremdir_b
                                               ,devlag    = developmentlag
                                               ,cumloss   = cumpaidloss_b
                                               ,lossratio = cumpaidloss_b / earnedpremdir_b
                                                )]

cohort_dt <- multi_dt[, .(maxtime = max(devlag), premium = unique(premium)), by = .(grcode, accyear)]
cohort_dt[, cohort_id := .I]
##      grcode accyear maxtime  premium cohort_id
##   1:     43    1988      10      957         1
##   2:     43    1989       9     3695         2
##   3:     43    1990       8     6138         3
##   4:     43    1991       7    17533         4
##   5:     43    1992       6    29341         5
##  ---                                          
## 146:   1767    1993       5 12533746       146
## 147:   1767    1994       4 13590797       147
## 148:   1767    1995       3 14401255       148
## 149:   1767    1996       2 14900682       149
## 150:   1767    1997       1 15065713       150
lst_standata <- list(growthmodel_id = 1   # Use weibull
                    ,n_data         = multi_dt[, .N]
                    ,n_time         = multi_dt[, length(unique(devlag))]
                    ,n_cohort       = cohort_dt[, length(unique(accyear))]
                    ,n_org          = cohort_dt[, length(unique(grcode))]
                    ,n_cohortdata   = cohort_dt[, .N]
                    ,cohort_id      = match(multi_dt$accyear, unique(cohort_dt$accyear))
                    ,org_id         = match(multi_dt$grcode, unique(cohort_dt$grcode))
                    ,t_value        = multi_dt[, sort(unique(devlag))]
                    ,t_idx          = multi_dt[, match(devlag, sort(unique(devlag)))]
                    ,premium        = multi_dt$premium
                    ,loss           = multi_dt$cumloss
                    ,cohort_maxtime = cohort_dt$maxtime
                    )

mi_2_stanmodel <- stan_model(stanfile, verbose = TRUE)
## 
## TRANSLATING MODEL 'losscurves_multiple' FROM Stan CODE TO C++ CODE NOW.
## successful in parsing the Stan model 'losscurves_multiple'.
mi_2_stanvb  <- vb(mi_2_stanmodel
                  ,data = lst_standata
                   )
## 
## This is Automatic Differentiation Variational Inference.
## 
## (EXPERIMENTAL ALGORITHM: expect frequent updates to the procedure.)
## 
## Gradient evaluation took 0.000356 seconds
## 1000 iterations under these settings should take 0.356 seconds.
## Adjust your expectations accordingly!
## 
## Begin eta adaptation.
## Iteration:   1 / 250 [  0%]  (Adaptation)
## Iteration:  50 / 250 [ 20%]  (Adaptation)
## Iteration: 100 / 250 [ 40%]  (Adaptation)
## Iteration: 150 / 250 [ 60%]  (Adaptation)
## Iteration: 200 / 250 [ 80%]  (Adaptation)
## Iteration: 250 / 250 [100%]  (Adaptation)
## Success! Found best value [eta = 0.1].
## 
## Begin stochastic gradient ascent.
##   iter       ELBO   delta_ELBO_mean   delta_ELBO_med   notes 
##    100     -2e+04             1.000            1.000
##    200     -1e+04             0.778            1.000
##    300     -1e+04             0.568            0.555
##    400     -1e+04             0.444            0.555
##    500     -1e+04             0.360            0.149
##    600     -9e+03             0.305            0.149
##    700     -9e+03             0.264            0.069
##    800     -9e+03             0.233            0.069
##    900     -9e+03             0.209            0.028
##   1000     -9e+03             0.189            0.028
##   1100     -9e+03             0.090            0.028
##   1200     -9e+03             0.036            0.020
##   1300     -9e+03             0.022            0.016
##   1400     -9e+03             0.015            0.014
##   1500     -8e+03             0.013            0.011
##   1600     -8e+03             0.011            0.011
##   1700     -8e+03             0.010            0.010
##   1800     -8e+03             0.009            0.009   MEAN ELBO CONVERGED   MEDIAN ELBO CONVERGED
## 
## Drawing a sample of size 1000 from the approximate posterior... 
## COMPLETED.
mi_2_stanfit <- sampling(mi_2_stanmodel
                        ,data    = lst_standata
                        ,iter    = 500
                        ,chains  = 8
                        ,verbose = TRUE
                         )
## 
## CHECKING DATA AND PREPROCESSING FOR MODEL 'losscurves_multiple' NOW.
## 
## COMPILING MODEL 'losscurves_multiple' NOW.
## 
## STARTING SAMPLER FOR MODEL 'losscurves_multiple' NOW.
mi_2_draws   <- extract(mi_2_stanfit, permuted = FALSE, inc_warmup = TRUE)
mi_2_monitor <- as.data.frame(monitor(mi_2_draws
                                     ,probs = c(0.025, 0.1, 0.25, 0.5, 0.75, 0.9, 0.975)
                                     ,print = FALSE))
mi_2_monitor$Parameter <- as.factor(gsub("\\[.*]", "", rownames(mi_2_monitor)))


plotdata_dt <- mi_2_monitor[, c('mean', '10%', '50%', '90%')]

setDT(plotdata_dt)
plotdata_dt[, variable := factor(rownames(mi_2_monitor)
                                ,levels = rownames(mi_2_monitor))]
##              mean         10%         50%         90%  variable
##    1:     1.44414     1.39334     1.44251     1.49421  omega[1]
##    2:     1.38916     1.29145     1.38680     1.49091  omega[2]
##    3:     1.25634     1.20762     1.25482     1.30623  omega[3]
##    4:     1.29241     1.23350     1.29031     1.35451  omega[4]
##    5:     1.32006     1.25360     1.31828     1.38868  omega[5]
##   ---                                                          
## 1919:     4.00000     4.00000     4.00000     4.00000 t_098[12]
## 1920:     5.00050     5.00000     5.00000     5.00000 t_098[13]
## 1921:     3.18100     3.00000     3.00000     4.00000 t_098[14]
## 1922:     4.98300     5.00000     5.00000     5.00000 t_098[15]
## 1923: -5460.77977 -5477.42885 -5460.37319 -5445.21673      lp__
ggplot(plotdata_dt[grep("hyper|sd_LR|_exp", variable)]) +
      geom_point(aes(x = variable, y = mean)) +
      geom_errorbar(aes(x = variable, ymin = `10%`, ymax = `90%`), width = 0) +
      expand_limits(y = 0) +
      theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
      xlab("Parameter") +
      ylab("Value")

We want to look at the multiple carriers expected LR is by themselves

ggplot(plotdata_dt[grep("mu_LR_exp", variable)]) +
      geom_point(aes(x = variable, y = mean)) +
      geom_errorbar(aes(x = variable, ymin = `10%`, ymax = `90%`), width = 0) +
      expand_limits(y = 0) +
      theme(axis.text.x = element_text(angle = 90, vjust = 0.5)) +
      xlab("Parameter") +
      ylab("Value")

5 References